Your instinct is right. I think that a native English speaker would say something like this: "I'm sorry, sir. I have just checked, and there are unfortunately no available tables for tonight". "Upon checking" is a correct expression in itself, but it isn't an appropriate phrase for this situation. It doesn't even fit grammatically with the rest of the sentence. ( For the whole utterance to be correct grammatically, the main part of the sentence would need to have the same subject as the gerund form 'checking'). This sounds like something a non-native might drop into their speech when making an effort to appear formal. It's not right. Don't use it.

Hi Jane, this is in fact the correct use of the word "upon" in this context. It's always the word "upon" + a verb ending in "ing", such as: Upon confirming, upon calling, upon further inspection.... all of these are perfectly correct and sound professional when dealing with clients.

Isn't it interesting having to have two different responses from two native speakers

This is the correct use of the word.